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Wave Length that gives minimum velocity?

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data
    The velocity of a wave of length L in deep water is v = K square root of (L/C + C/L)
    where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?

    2. Relevant equations
    Possibly a(t) = v'(t) = s"(t)

    3. The attempt at a solution
    I don't know how to work the question but here is my best guess...
    Can I just cancel L with L and C with C (inside the square root) and be left with v = K.
    Next, to find the length, can I just find the antiderivative of the velocity, like this: v(t) = Kt + C.

    How do I workout this question?
     
  2. jcsd
  3. Mar 23, 2010 #2

    tiny-tim

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    Hi 01010011! Welcome to PF! :smile:

    The question is asking for the minimum of (L/C + C/L), where C is a constant. :wink:
     
  4. Mar 23, 2010 #3
    Thanks for the welcome. Well I was thinking the answer was 0 because the Ls and Cs cancel out each other, but Im not sure
     
  5. Mar 23, 2010 #4

    tiny-tim

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    They don't cancel (why do you think they would? :confused:)

    Try differentiating. :smile:
     
  6. Mar 23, 2010 #5
    Differentiating v = K square root of (L/C + C/L)? I thought I could just cancel the Ls and Cs. Here is another attempt:

    v = k square root (L/C + C/L)

    v(t) = s'(t)

    v(t) = k square root [(L*C^-1) + (C*L^-1)] + C

    v(t) = k * [(L*C^-1) + (C*L^-1)] ^ (1/2) + C

    s(t) = k * {[(L*C^-1) + (C*L^-1)] ^ (3/2)} / 3/2 + C + D

    This looks like madness. Im sure this is not correct.
    What steps (and why) should I take to answer questions like this?
     
  7. Mar 24, 2010 #6
    Right, since C is a constant, the smallest value C can be is 0, So....
     
  8. Mar 24, 2010 #7

    tiny-tim

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    Nooo, this is a mess. :redface:

    i] s has nothing to do with it, the fact that v is a velocity has nothing to do with it, all you have to do is find the minimum of (L/C + C/L), where C is a constant

    ii] why are you trying to minimise the square-root? the square-root is a minimum if and only if the whole thing is a minimum, so minimise the whole thing, it's easier! :wink:

    iii] you haven't used the chain rule at all … look it up in your book :smile:

    erm :redface: … that doesn't even make sense, does it?

    get some sleep! :zzz:​
     
  9. Mar 24, 2010 #8
    Ok, I got some much needed sleep lol!

    Alright, let me try again:

    v = K square root of (L/C + C/L)
    dy/dx?
    Let U = square root of (L/C + C/L)
    V = KU
    V = 1KU^(1-1)
    V = K

    hmmm....
     
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